where a spin\(^c\)  structure \(\mathfrak {s}\)  is self-conjugate if it is isomorphic to its conjugate \(\mathfrak {s}\cong \bar{\mathfrak {s}}\) . Equivalently, \(\mathfrak {s}\)  is induced by some spin structure (as \(b_1(M_{\varphi })=1\) , it is induced by exactly two of them). We will denote the spin\(^c\)  structure corresponding to \(L\)  by \(\mathfrak {s}_L\) . This is closely related to the classical observation due to Atiyah [1]} that there is a natural bijection
\(\lbrace \text{spin structures on $\Sigma $}\rbrace \longleftrightarrow \lbrace \text{theta characteristics }L\rbrace .\) 
